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<p>This paper investigates a nonlinear stochastic discrete modified Swift-Hohenberg (S-H) equation driven by superlinear noise. We first demonstrate the global existence and uniqueness of solutions provided the noise coefficient exhibits a superlinear growth. Then, we establish the existence of a unique mean random attractor for the non-autonomous system in the Bochner space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( L^2(\Omega , \ell ^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, under appropriate conditions on the drift and diffusion terms, we prove that the autonomous system admits a unique invariant probability measure in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ell ^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, characterized by ergodicity, exponential mixing, and Lyapunov stability. To address the challenges posed by non-compactness in infinite lattices and the infinite-dimensional nature of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \ell ^2 \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, we implement uniform tail-estimation strategies to verify the tightness of solution distribution laws.</p>

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Existence, Ergodicity and Exponential Mixing of Invariant Measures for Modified Swift-Hohenberg Lattice Systems Driven by Superlinear Noise

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Wei Feng,
  • Pengyu Chen,
  • Yongxiang Li

摘要

This paper investigates a nonlinear stochastic discrete modified Swift-Hohenberg (S-H) equation driven by superlinear noise. We first demonstrate the global existence and uniqueness of solutions provided the noise coefficient exhibits a superlinear growth. Then, we establish the existence of a unique mean random attractor for the non-autonomous system in the Bochner space \( L^2(\Omega , \ell ^2)\) L 2 ( Ω , ℓ 2 ) . Furthermore, under appropriate conditions on the drift and diffusion terms, we prove that the autonomous system admits a unique invariant probability measure in \(\ell ^2\) ℓ 2 , characterized by ergodicity, exponential mixing, and Lyapunov stability. To address the challenges posed by non-compactness in infinite lattices and the infinite-dimensional nature of \( \ell ^2 \) ℓ 2 , we implement uniform tail-estimation strategies to verify the tightness of solution distribution laws.